Mathematics Asked by Aulwmate on October 5, 2020

I have approximate the second order derivative of some function $f(x)$ on some interval $[a,b]$. Let $ h =frac{b-a}{n}$ for some positive integer $n$, the discretization of $[a, b]$ will be the following:

$$

a=x_0,\

x_1 = x_2 +h\

x_2 = x_0 +2h\

x_3 = x_0 +3h\

vdots \

x_n = x_0 +nh = b.

$$

For $i = 2, 3,ldots, n-1$, the second order approximation of $f(x)$ will

$$

f^{”}(x_i) approx frac{f(x_i+h)-2f(x_i)+f(x_i-h)}{h^2}

$$

But if I try to use the same formula to approximate the second order derivative at $x_0$ and $x_n$, I need the values of $f$ at $x_0-h$ and $x_n+h$.

Is there any other way to approximate $f^{”}$ at the end points?

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